## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 47, Number 4 (2015), 1190-1211.

### Escape from the boundary in Markov population processes

A. D. Barbour, K. Hamza, Haya Kaspi, and F. C. Klebaner

#### Abstract

Density dependent Markov population processes in large populations of size
*N* were shown by Kurtz (1970), (1971) to be well approximated over finite
time intervals by the solution of the differential equations that describe
their average drift, and to exhibit stochastic fluctuations about this
deterministic solution on the scale √*N* that can be approximated
by a diffusion process. Here, motivated by an example from evolutionary
biology, we are concerned with describing how such a process leaves an
absorbing boundary. Initially, one or more of the populations is of size much
smaller than *N*, and the length of time taken until all populations have
sizes comparable to *N* then becomes infinite as
*N* → ∞. Under suitable assumptions, we show that in the
early stages of development, up to the time when all populations have sizes at
least *N*^{1-α} for 1/3 < α < 1, the
process can be accurately approximated in total variation by a Markov branching
process. Thereafter, it is well approximated by the deterministic solution
starting from the original initial point, but with a random time delay.
Analogous behaviour is also established for a Markov process approaching an
equilibrium on a boundary, where one or more of the populations become extinct.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 47, Number 4 (2015), 1190-1211.

**Dates**

First available in Project Euclid: 11 December 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1449859806

**Digital Object Identifier**

doi:10.1239/aap/1449859806

**Mathematical Reviews number (MathSciNet)**

MR3433302

**Zentralblatt MATH identifier**

1355.92085

**Subjects**

Primary: 92D30: Epidemiology

Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

**Keywords**

Markov population process boundary behaviour branching process

#### Citation

Barbour, A. D.; Hamza, K.; Kaspi, Haya; Klebaner, F. C. Escape from the boundary in Markov population processes. Adv. in Appl. Probab. 47 (2015), no. 4, 1190--1211. doi:10.1239/aap/1449859806. https://projecteuclid.org/euclid.aap/1449859806